Uniqueness Theorems of Difference Operator on Entire Functions

A function f(z) is called meromorphic, if it is analytic in the complex plane except at poles. It is assumed that the reader is familiar with the standard symbols and fundamental results of Nevanlinna theory such as the characteristic function T(r, f), and proximity function m(r, f), counting function N(r, f) (see [1, 2]). In addition we use S(r, f) denotes any quantity that satisfies the condition: S(r, f) = o(T(r, f)) as r → ∞ possibly outside an exceptional set of finite logarithmic measure. Let f and g be two nonconstant meromorphic functions, a ∈ C ∪ {∞}, we say that f and g share the value a IM (ignoringmultiplicities) iff−a and g−a have the same zeros, they share the value a CM (counting multiplicities) if f − a and g − a have the same zeros with the same multiplicities. When a = ∞ the zeros of f−amean the poles of f (see [2]). Let p be a positive integer and a ∈ C ∪ {∞}. We use Np)(r, 1/(f − a)) to denote the counting function of the zeros of f − a (counted with proper multiplicities) whose multiplicities are not bigger than p, N(p+1(r, 1/(f − a)) to denote the counting function of the zeros of f − a whose multiplicities are not less than p + 1. Np)(r, 1/(f − a)) and N(p+1(r, 1/(f − a)) denote their corresponding reduced counting functions (ignoring multiplicities), respectively. We denote by E(a, f) the set of zeros of f − a with multiplicity, Ep)(a, f) the set of zeros of f − a (counted with proper multiplicities) whose multiplicities are not greater than p. In 1997, Yang and Hua (see [3]) studied the uniqueness of the differential monomials and obtained the following result.